Regarding first paragraph, you seem to be fully right. Because, as load becomes bigger, the problem is starting, i.e. with a led as load no problem, with 0,5A lamp as the load the problem is there, with 10W lamp worse. So, I separated the circuit power and load power feed, the problem seems to be solved. Thanks for your idea.

When building it is a good idea to separate blocks and test them separately. I always test with a lab power supply and later add the specific power supply.

On the other hand, I couldn't under stand your proposal about voltage dividers. Can you say it schematically? Current demand from relay contacts are about 1-1,2A DC at 12V.

If you have a voltage between 6 and 12 you can put a 6 V load between this voltage and the 12 V rail. Another possibility is to do something like this:

 

When building it is a good idea to separate blocks and test them separately. I always test with a lab power supply and later add the specific power supply.
If you have a voltage between 6 and 12 you can put a 6 V load between this voltage and the 12 V rail. Another possibility is to do something like this:

View attachment 93114

I suppose that diode is a led and together with 560 ohm resistor it represents the load. And also resistor values are given as ohms. Pls. confirm.